Derivative of a Polynomial Function
Derivative of a Polynomial Function. Practice questions to deepen understanding of the derivative of a polynomial function. Online math practice with full solutions and step-by-step explanations.
Derivative of a Polynomial Function — basic differentiation rules, derivative of x^n, sum and difference. Comprehensive practice with explanations.
Given \(f(x)=3x^2+5x-7\). What is \(f\''(x)\)?
Solution: Power rule: \(3x^2\to6x\), \(5x\to5\), constant \(\to0\).
What is the derivative of \(f(x)=4x^3\)?
Solution: \(4x^3\to4\cdot3x^2=12x^2\)
Given \(f(x)=-2x^2+8\). Find \(f\''(x)\).
Solution: \(-2x^2\to-4x\), constant \(\to0\)
Differentiate \(f(x)=5x^4-3x^2+x-1\).
Solution: Term by term: \(20x^3-6x+1\).
What is the derivative of \(f(x)=\frac{{1}}{{2}}x^2+3x\)?
Solution: \(\frac{1}{2}x^2\to x\), \(3x\to3\)
The derivative of \(f(x)=7x^3-x\) is:
Solution: \(7x^3\to21x^2\), \(-x\to-1\)
What is the derivative of \(f(x)=x^5\)?
Solution: Power rule: \(x^5\to5x^4\)
Given \(f(x)=-3x^4+2x^3-x+9\). Find the derivative.
Solution: Term by term: \(-12x^3+6x^2-1\). Constant disappears.
What is the derivative of \(f(x)=\frac{{2}}{{3}}x^3\)?
Solution: \(\frac{2}{3}\cdot3x^2=2x^2\)
Given \(f(x)=6x^2+4x+10\). What is the derivative?
Solution: \(6x^2\to12x\), \(4x\to4\), \(10\to0\)
Differentiate \(f(x)=x^3-5x^2+4x\).
Solution: \(3x^2-10x+4\)
What is the derivative of \(f(x)=-x^3+3x\)?
Solution: \(-x^3\to-3x^2\), \(3x\to3\)
Given \(f(x)=2x^4+x^2\). Find \(f\''(x)\).
Solution: \(2x^4\to8x^3\), \(x^2\to2x\)
What is the derivative of \(f(x)=\frac{{1}}{{4}}x^4-x\)?
Solution: \(\frac{1}{4}x^4\to x^3\), \(-x\to-1\)
The derivative of \(f(x)=9x-7\) is:
Solution: Linear function: derivative = coefficient = 9.
What is the derivative of \(f(x)=5-2x\)?
Solution: \(\frac{d}{dx}(5)=0\), \(\frac{d}{dx}(-2x)=-2\)
Differentiate \(f(x)=x^6\).
Solution: Power rule: \(x^6\to6x^5\)
Given \(f(x)=3x^5-x^3+2x\). What is the derivative?
Solution: Term by term: \(15x^4-3x^2+2\)
What is the derivative of \(f(x)=-4x^3-6x\)?
Solution: \(-4x^3\to-12x^2\), \(-6x\to-6\)
Given \(f(x)=0.5x^2-4x+1\). What is the derivative?
Solution: \(0.5\cdot2x=x\), derivative of \(-4x\) is \(-4\)
Differentiate \(f(x)=x^4-4x^2\).
Solution: \(x^4\to4x^3\), \(-4x^2\to-8x\)
Given \(f(x)=2x^3+3x^2-x+5\). What is the derivative?
Solution: Term by term: \(6x^2+6x-1\)
What is the derivative of \(f(x)=-x^4+4x\)?
Solution: \(-x^4\to-4x^3\), \(4x\to4\)
Given \(f(x)=7x^2+\frac{{1}}{{3}}x^3\). What is the derivative?
Solution: \(7x^2\to14x\), \(\frac{1}{3}x^3\to x^2\)
What is the derivative of \(f(x)=\frac{{5}}{{2}}x^2\)?
Solution: \(\frac{5}{2}\cdot2x=5x\)
Given \(f(x)=-\frac{{1}}{{2}}x^2+x^3\). What is the derivative?
Solution: \(-\frac{1}{2}x^2\to-x\), \(x^3\to3x^2\)
What is the derivative of \(f(x)=4x^4-8x+3\)?
Solution: \(4x^4\to16x^3\), \(-8x\to-8\), \(3\to0\)
Given \(f(x)=x^3+6\). What is the derivative?
Solution: \(x^3\to3x^2\), constant \(\to0\)
What is the derivative of \(f(x)=10x^2-x\)?
Solution: \(10x^2\to20x\), \(-x\to-1\)
Given \(f(x)=-2x^5+x^2\). Find the derivative.
Solution: \(-2x^5\to-10x^4\), \(x^2\to2x\)